Subspace method for local trend extraction

A time series is usually decomposed as trend and irregular parts. Generally the trend part is treated by regression or filtering methods. There are some shortcomings associated with these methods, either the form of trend is too simple to represent some complex trend patterns, or no concrete trend formula is available. Considering the trend part as the inherent dynamics conveyed by the time series, the long time behavior is determined by the trend dynamics, a succinct trend representation is thus meaningful and useful. We use systems theory to extract trend pattern in state space form from a time series. The resultant state space model is flexible enough to accommodate complex trend patterns. The computation needed for state space model building is singular value decomposition (SVD) for a data matrix, a reliable algorithm with good numerical property. The merit of the state space model is verified by a numerical simulation.